A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting $k$-body fermionic Embedded Gaussian Ensembles

Abstract

In spite of its simplicity, the central limit theorem captures one of the most outstanding phenomena in mathematical physics, that of universality. It states that, while navigating the set of all possible distributions by their convolution there exists a fixed point, the family of Gaussian distributions, with a fairly large basin of attraction comprising a wide family of weakly correlated distributions. More colloquially, this means roughly speaking that the sum of independent and identically distributed random variables follows a Gaussian distribution irrespective of the details of the original distributions from which these random variables are drawn. While this classical result is well understood, and is the reason behind many physical phenomena, it is still not very clear what happens to this universal behaviour when the random variables become correlated. Not a general overarching theorem exists on a possible emergence of new universal behaviour and, so far as we are aware of, this area of research must be explored in a case-by-case basis. A fruitful mathematically laboratory to investigate the rising of new universal properties is offered by the set of eigenvalues of random matrices. In this regard a lot of work has been done using the standard random matrix ensembles and focusing on the distribution of extreme eigenvalues. In this case, the distribution of the largest -or smallest-eigenvalue departs from the Fisher-Tippett-Gnedenko theorem yielding the celebrated Tracy-Widom distribution. One may wonder, yet again, how robust is this new universal behaviour captured by the Tracy-Widom distribution when the correlation among eigenvalues changes. Few answers have been provided to this poignant question and our intention in the present work is to contribute to this interesting unexplored territory. Thus, we study numerically the probability distribution for the normalized largest eigenvalue of the interacting k-body fermionic orthogonal and unitary Embedded Gaussian Ensembles in the diluted limit. We find a smooth transition from a slightly asymmetric Gaussian-like distribution, for small k/m, to the Tracy-Widom distribution as k/m → 1, where k is the rank of the interaction and m is the number of fermions. Correlations at the edge of the spectrum are stronger for small values of k/m, and are independent of the number of particles considered. Our results indicate that subtle correlations towards the edge of the spectrum distinguish the statistical properties of the spectrum of interacting many-body systems in the dilute limit, from those expected for the standard random matrix ensembles.